Open Access

Existence results for classes of infinite semipositone problems

  • Jerome GoddardII1,
  • Eun Kyoung Lee2,
  • Lakshmi Sankar3 and
  • R Shivaji4Email author
Boundary Value Problems20132013:97

DOI: 10.1186/1687-2770-2013-97

Received: 23 October 2012

Accepted: 5 April 2013

Published: 19 April 2013

Abstract

We consider the problem

{ Δ p u = a u p 1 b u γ 1 c u α , x Ω , u = 0 , x Ω ,

where Δ p u = div ( | u | p 2 u ) , p > 1 , Ω is a smooth bounded domain in R n , a > 0 , b > 0 , c 0 , γ > p and α ( 0 , 1 ) . Given a, b, γ and α, we establish the existence of a positive solution for small values of c. These results are also extended to corresponding exterior domain problems. Also, a bifurcation result for the case c = 0 is presented.

1 Introduction

Consider the nonsingular boundary value problem:
{ Δ u = a u b u 2 c h ( x ) , x Ω , u = 0 , x Ω ,
(1)

where Ω is a smooth bounded domain in R n , a > 0 , b > 0 , c 0 , Δ u = div ( u ) is the Laplacian of u and h : Ω ¯ R is a C 1 ( Ω ¯ ) function satisfying h ( x ) 0 for x Ω , h ( x ) 0 , max x Ω ¯ h ( x ) = 1 and h ( x ) = 0 for x Ω . Existence of positive solutions of problem (1) was studied in [1]. In particular, it was proved that given an a > λ 1 and b > 0 there exists a c ( a , b , Ω ) > 0 such that for c < c (1) has positive solutions. Here, λ 1 is the first eigenvalue of −Δ with Dirichlet boundary conditions. Nonexistence of a positive solution was also proved when a λ 1 . Later in [2], these results were extended to the case of the p-Laplacian operator, Δ p , where Δ p u = div ( | u | p 2 u ) , p > 1 . Boundary value problems of the form (1) are known as semipositone problems since the nonlinearity f ( s , x ) = a s b s 2 c h ( x ) satisfies f ( 0 , x ) < 0 for some x Ω . See [39] for some existence results for semipositone problems.

In this paper, we study positive solutions to the singular boundary value problem:
{ Δ p u = a u p 1 b u γ 1 c u α , x Ω , u = 0 , x Ω ,
(2)

where Δ p u = div ( | u | p 2 u ) , p > 1 , Ω is a smooth bounded domain in R n , a > 0 , b > 0 , c 0 , α ( 0 , 1 ) , p > 1 , and γ > p . In the literature, problems of the form (2) are referred to as infinite semipositone problems as the nonlinearity f ( s ) = a s p 1 b s γ 1 c s α satisfies lim s 0 + f ( s ) = . One can refer to [1014], and [1517] for some recent existence results of infinite semipositone problems. We establish the following theorem.

Theorem 1.1 Given a , b > 0 , γ > p , and α ( 0 , 1 ) , there exists a constant c 1 = c 1 ( a , b , α , p , γ , Ω ) > 0 such that for c < c 1 , (2) has a positive solution.

Remark 1.1 In the nonsingular case ( α = 0 ), positive solutions exist only when a > λ 1 (the principal eigenvalue) (see [1, 2]). But in the singular case, we establish the existence of a positive solution for any given a > 0 .

Next, we study positive radial solutions to the problem:
{ Δ p u = K ( | x | ) ( a u p 1 b u γ 1 c u α ) , x Ω , u = 0 , if  | x | = r 0 , u 0 , as  | x | ,
(3)
where Ω = { x R n | | x | > r 0 } is an exterior domain, n > p , a > 0 , b > 0 , c 0 , α ( 0 , 1 ) , p > 1 , γ > p and K : [ r 0 , ) ( 0 , ) belongs to a class of continuous functions such that lim r K ( r ) = 0 . By using the transformation: r = | x | and s = ( r r 0 ) n + p p 1 , we reduce (3) to the following boundary value problem:
{ ( | u | p 2 u ) = h ( s ) ( a u p 1 b u γ 1 c u α ) , 0 < s < 1 , u ( 0 ) = u ( 1 ) = 0 ,
(4)

where h ( s ) = ( p 1 n p ) p r 0 p s p ( n 1 ) n p K ( r 0 s ( p 1 ) n p ) . We assume:

( H 1 ) K C ( [ r 0 , ) , ( 0 , ) ) and satisfies K ( r ) < 1 r n + θ for r 1 , and for some θ such that ( n p p 1 ) α < θ < n p p 1 .

With the condition ( H 1 ), h satisfies:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ5_HTML.gif
(5)

We note that if θ n p p 1 then h ( s ) is nonsingular at 0 and h C ( [ 0 , 1 ] , ( 0 , ) ) . In this case, problem (4) can be studied using ideas in the proof of Theorem 1.1. Hence, our focus is on the case when θ < n p p 1 in which, h may be singular at 0. Note that in this case h ˆ = inf s ( 0 , 1 ) h ( s ) > 0 .

Remark 1.2 Note that ρ + α < 1 since θ > ( n p p 1 ) α .

We then establish the following theorem.

Theorem 1.2 Given a , b > 0 , γ > p , α ( 0 , 1 ) , and assume ( H 1 ) holds. Then there exists a constant c 2 = c 2 ( a , b , α , p , γ ) > 0 such that for c < c 2 , (3) has a positive radial solution.

Finally, we prove a bifurcation result for the problem
{ Δ p u = a u p 1 b u γ 1 u α , x Ω , u = 0 , on  Ω ,
(6)

where Ω is a smooth bounded domain in R n , a is a positive parameter, b , α > 0 , p > 1 + α and γ > p . We prove the following.

Theorem 1.3 The boundary value problem (6) has a branch of positive solutions bifurcating from the trivial branch of solutions ( a , 0 ) at ( 0 , 0 ) (as shown in Figure 1).
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig1_HTML.jpg
Figure 1

Bifurcation diagram, a vs. u for ( 6 ).

Our results are obtained via the method of sub-super solutions. By a subsolution of (2), we mean a function ψ W 1 , p ( Ω ) C ( Ω ¯ ) that satisfies
{ Ω | ψ | p 2 ψ w d x Ω a ψ p 1 b ψ γ 1 c ψ α w d x , for every  w W , ψ > 0 , in  Ω , ψ = 0 , on  Ω ,
and by a supersolution we mean a function Z W 1 , p ( Ω ) C ( Ω ¯ ) that satisfies:
{ Ω | Z | p 2 Z w d x Ω a Z p 1 b Z γ 1 c Z α w d x , for every  w W , Z > 0 , in  Ω , Z = 0 , on  Ω ,

where W = { ξ C 0 ( Ω ) | ξ 0  in  Ω } . The following lemma was established in [13].

Lemma 1.4 (see [13, 18])

Let ψ be a subsolution of (2) and Z be a supersolution of (2) such that ψ Z in Ω. Then (2) has a solution u such that ψ u Z in Ω.

Finding a positive subsolution, ψ, for such infinite semipositone problems is quite challenging since we need to construct ψ in such a way that lim x Ω Δ p ψ = and Δ p ψ > 0 in a large part of the interior. In this paper, we achieve this by constructing subsolutions of the form ψ = k ϕ 1 β , where k is an appropriate positive constant, β ( 1 , p p 1 ) and ϕ 1 is the eigenfunction corresponding to the first eigenvalue of Δ p ϕ = λ | ϕ | p 2 ϕ in Ω, ϕ = 0 on Ω.

In Sections 2, 3, and 4, we provide proofs of our results. Section 5 is concerned with providing some exact bifurcation diagrams of positive solutions of (2) when Ω = ( 0 , 1 ) and p = 2 .

2 Proof of Theorem 1.1

We first construct a subsolution. Consider the eigenvalue problem Δ p ϕ = λ | ϕ | p 2 ϕ in Ω, ϕ = 0 on Ω. Let ϕ 1 be an eigenfunction corresponding to the first eigenvalue λ 1 such that ϕ 1 > 0 and ϕ 1 = 1 . Also, let δ , m , μ > 0 be such that | ϕ 1 | m in Ω δ and ϕ 1 μ in Ω Ω δ , where Ω δ = { x Ω | d ( x , Ω ) δ } . Let β ( 1 , p p 1 + α ) be fixed. Here, note that since α ( 0 , 1 ) , p p 1 + α > 1 . Choose a k > 0 such that 2 b k γ p + β p 1 λ 1 k α a . Define c 1 = min { k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p , 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α ) } . Note that c 1 > 0 by the choice of k and β. Let ψ = k ϕ 1 β . Then
Δ p ψ = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) .
To prove ψ is a subsolution, we need to establish:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ7_HTML.gif
(7)
in Ω if c < c 1 . To achieve this, we split the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) into three, namely,
k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = a k p 1 α ϕ 1 β ( p 1 α ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) .
Now to prove (7) holds in Ω, it is enough to show the following three inequalities:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ8_HTML.gif
(8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ9_HTML.gif
(9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ10_HTML.gif
(10)
From the choice of k, ( a β p 1 λ 1 k α ) 2 b k γ p , hence,
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) .
(11)
Using ϕ 1 μ in Ω Ω δ and c < 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α )
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) k p 1 ϕ 1 β ( p 1 ) ( a k α λ 1 β p 1 ) 2 k α ϕ 1 α β c k α ϕ 1 α β .
(12)
Finally, since | ϕ 1 | m , in Ω δ , and c < k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p ,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β ϕ 1 p β ( p 1 ) α β c k α ϕ 1 α β ϕ 1 p β ( p 1 + α ) .
Since p β ( p 1 + α ) > 0 ,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) c k α ϕ 1 α β .
(13)

From (11), (12) and (13) we see that equation (7) holds in Ω, if c < c 1 . Next, we construct a supersolution. Let e be the solution of Δ p e = 1 in Ω , e = 0 on Ω. Choose M ¯ > 0 such that a u p 1 b u γ 1 c u α M ¯ p 1 u > 0 and M ¯ e ψ . Define Z = M ¯ e . Then Z is a supersolution of (2). Thus, Theorem 1.1 is proven.

3 Proof of Theorem 1.2

We begin the proof by constructing a subsolution. Consider
( | ϕ | p 2 ϕ ) = λ | ϕ | p 2 ϕ , t ( 0 , 1 ) , ϕ ( 0 ) = ϕ ( 1 ) = 0 .
(14)
Let ϕ 1 be an eigenfunction corresponding to the first eigenvalue of (14) such that ϕ 1 > 0 and ϕ 1 = 1 . Then there exist d 1 > 0 such that 0 < ϕ 1 ( t ) d 1 t ( 1 t ) for t ( 0 , 1 ) . Also, let ϵ < ϵ 1 and m , μ > 0 be such that | ϕ 1 | m in ( 0 , ϵ ] [ 1 ϵ , 1 ) and ϕ 1 μ in ( ϵ , 1 ϵ ) . Let β ( 1 , p ρ p 1 + α ) be fixed and choose k > 0 such that 2 b k γ p + β p 1 λ 1 k α h ˆ a . Define c 2 = min { k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p d 1 ρ , 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α h ˆ ) } . Then c 2 > 0 by the choice of k and β. Let ψ = k ϕ 1 β . This implies that:
( | ψ | p 2 ψ ) = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) .
To prove ψ is a subsolution, we need to establish:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ15_HTML.gif
(15)
Here, we note that the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = h ˆ k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) h ˆ h ( t ) ( a k p 1 α ϕ 1 β ( p 1 α ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) 1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) ) , where h ˆ = inf s ( 0 , 1 ) h ( s ) . Now to prove (15) holds in ( 0 , 1 ) , it is enough to show the following three inequalities:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ16_HTML.gif
(16)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ17_HTML.gif
(17)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Equ18_HTML.gif
(18)
From the choice of k, ( a β p 1 λ 1 k α h ˆ ) 2 b k γ p , hence,
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) .
(19)
Using ϕ 1 μ in ( ϵ , 1 ϵ ) and c < 1 2 k p 1 μ β ( p 1 ) ( a β p 1 λ 1 k α h ˆ )
1 2 k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 h ˆ ) k p 1 ϕ 1 β ( p 1 ) ( a k α λ 1 β p 1 h ˆ ) 2 k α ϕ 1 α β c k α ϕ 1 α β .
(20)
Next, we prove (18) holds in ( 0 , ϵ ] . Since | ϕ 1 | m , and p β ( p 1 ) > α β + ρ
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β ϕ 1 ρ k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p k α ϕ 1 α β d 1 ρ t ρ .
Since h ( t ) 1 t ρ in ( 0 , ϵ ] , and c < k p 1 + α β p 1 ( β 1 ) ( p 1 ) m p d 1 ρ ,
k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) c h ( t ) k α ϕ 1 α β .
(21)

Proving (18) holds in [ 1 ϵ , 1 ) is straightforward since h is not singular at t = 1 . Thus, from equations (19), (20) and (21), we see that (15) holds in ( 0 , 1 ) . Hence, ψ is a subsolution. Let Z = M ¯ e where e satisfies ( | e | p 2 e ) = h ( t ) in ( 0 , 1 ) , e ( 0 ) = e ( 1 ) = 0 and M ¯ is such that a u p 1 b u γ 1 c u α M ¯ p 1 u > 0 and M ¯ e ψ . Then Z is a supersolution of (4) and there exists a solution u of (4) such that u [ ψ , Z ] . Thus, Theorem 1.2 is proven.

4 Proof of Theorem 1.3

We first prove (6) has a positive solution for every a > 0 . We begin by constructing a subsolution. Let ϕ 1 be as in the proof of Theorem 1.1 (see Section 2). Let β ( 1 , p p 1 ) , and choose a k > 0 such that b k γ p + β p 1 λ 1 k α a . Let ψ = k ϕ 1 β . Then
Δ p ψ = k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) k p 1 β p 1 ( β 1 ) ( p 1 ) | ϕ 1 | p ϕ 1 p β ( p 1 ) .
To prove ψ is a subsolution, we will establish:
k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) a k p 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α )
(22)
in Ω. To achieve this, we rewrite the term k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) as k p 1 β p 1 λ 1 ϕ 1 β ( p 1 ) = a k p 1 α ϕ 1 β ( p 1 α ) k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) . Now to prove (22) holds in Ω, it is enough to show k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( γ 1 α ) . From the choice of k, ( a β p 1 λ 1 k α ) b k γ p , hence,
k p 1 α ϕ 1 β ( p 1 α ) ( a k α ϕ 1 α β β p 1 λ 1 ) b k γ 1 α ϕ 1 β ( p 1 α ) b k γ 1 α ϕ 1 β ( γ 1 α ) .

Thus, ψ is a subsolution. It is easy to see that Z = ( a b ) 1 γ p is a supersolution of (6). Since k, can be chosen small enough, ψ Z . Thus, (6) has a positive solution for every a > 0 . Also, all positive solutions are bounded above by Z. Hence, when a is close to 0, every positive solution of (6) approaches 0. Also, u 0 is a solution for every a. This implies we have a branch of positive solutions bifurcating from the trivial branch of solutions ( a , 0 ) at ( 0 , 0 ) .

5 Numerical results

Consider the boundary value problem
{ u ( x ) = a u b u 2 c u α , x ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) ,
(23)
where a , b > 0 , c 0 and α ( 0 , 1 ) . Using the quadrature method (see [19]), the bifurcation diagram of positive solutions of (23) is given by
G ( ρ , c ) = 0 ρ d s [ 2 ( F ( ρ ) F ( s ) ) ] = 1 2 ,
(24)
where F ( s ) : = 0 s f ( t ) d t where f ( t ) = a t b t 2 c t α and ρ = u ( 1 2 ) = u . We plot the exact bifurcation diagram of positive solutions of (23) using Mathematica. Figure 2 shows bifurcation diagrams of positive solutions of (23) when a = 8 ( < λ 1 ) and b = 1 for different values of α.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig2_HTML.jpg
Figure 2

Bifurcation diagrams, c vs. ρ for ( 23 ) with a = 8 , b = 1 .

Bifurcation diagrams of positive solutions of (23) when a = 15 ( > λ 1 ) and b = 1 for different values of α is shown in Figure 3.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig3_HTML.jpg
Figure 3

Bifurcation diagrams, c vs. ρ for ( 23 ) with a = 15 , b = 1 .

Finally, we provide the exact bifurcation diagram for (6) when p = 2 , and Ω = ( 0 , 1 ) . Consider
{ u ( x ) = a u b u 2 u α , x ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) ,
(25)
where a , b , α > 0 . The bifurcation diagram of positive solutions of (25) is given by
G ˜ ( ρ , a ) = 0 ρ d s [ 2 ( F ˜ ( ρ ) F ˜ ( s ) ) ] = 1 2 ,
(26)
where F ˜ ( s ) : = 0 s f ˜ ( t ) d t where f ˜ ( t ) = a t b t 2 t α and ρ = u ( 1 2 ) = u . The bifurcation diagram of positive solutions of (25) as well as the trivial solution branch are shown in Figure 4 when α = 0.5 and b = 1 .
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-97/MediaObjects/13661_2012_Article_352_Fig4_HTML.jpg
Figure 4

Bifurcation diagram, a vs. ρ for ( 25 ) with α = 0.5 , b = 1 .

Declarations

Acknowledgements

EK Lee was supported by 2-year Research Grant of Pusan National University.

Authors’ Affiliations

(1)
Department of Mathematics, Auburn University Montgomery
(2)
Department of Mathematics Education, Pusan National University
(3)
Department of Mathematics & Statistics, Mississippi State University
(4)
Department of Mathematics & Statistics, University of North Carolina at Greensboro

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